There are no necklaces on Day 5 that you can't get before day 5, and some may even be inaccessible on Day 5. As a general rule, not much side stuff can be done on Day 5 including quests so it is recommended to get everything not related to the main story done before progressing to Day 5.

There are a total of 110 quests, and to get the Beta Tester to finish you'll need to leave the room and then re-enter and he should have fixed the bug (teleporting may not count).

Some skills may become untrainable on Day 5 so it would be best to train them all to 99 on Day 4 for that quest if you want to complete that quest. Also on Day 5 some townspeople will disappear so some quest givers may or may not be there on Day 5.